On the multiplicities of normalized Laplacian eigenvalues of graphs

Abstract

Let G be a simple graph of order n with normalized Laplacian eigenvalues ρ1≥ρ2≥⋯≥ρn−1≥ρn=0. Let mG(ρi)(1≤i≤n) be the multiplicity of the normalized Laplacian eigenvalue ρi of G. In this paper, we give some necessary and sufficient conditions for a connected graph G to have mG(ρi)=n−3 for some i. We also discuss about graphs with mG(ρn−2)=n−k. Moreover, we completely characterize the connected graphs with mG(ρn−1)=n−k for sufficiently large n.